Re: More strange behavior by ComplexExpand

• To: mathgroup at smc.vnet.net
• Subject: [mg60622] Re: [mg60603] More strange behavior by ComplexExpand
• From: Pratik Desai <pdesai1 at umbc.edu>
• Date: Thu, 22 Sep 2005 02:08:15 -0400 (EDT)
• References: <200509210720.DAA08138@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Raul Martinez wrote:

>To Mathgroup,
>
>I use Mathematica 5.2 with Mac OS X (Tiger).
>
>Add the following to a recent thread on the sometimes strange
>behavior of ComplexExpand.
>
>I used ComplexExpand with an argument in which all the variables in
>the argument of the function are real. Since ComplexExpand is
>supposed to assume that all variables are real by default, one would
>expect ComplexExpand to return the expression without change, but it
>doesn't.
>
This is not exactly true, in mathematica all the variables are assumed
complex (alteast what I have experienced so far)

In my opinion not only doyou have to specifically assign your variable
to be real by using Assuming or \$Assumptions +Simplify, but also you
have specifiy the values of "a" on the real line and hence your
expression will change based on the complex expand algorithm
Remove["Global`*"]

\$Assumptions = {-â?? < a < 0, t Ïµ Reals}
s1=ComplexExpand[(a/Ï?)^(1/4) *Exp[-(a t^2)/2]] // Simplify
>>((1 + I)*(a^2)^(1/8)*Sqrt[E^(a*t^2)])/(Sqrt[2]*E^(a*t^2)*Pi^(1/4))

\$Assumptions = {0 < a < â??, t Ïµ Reals}
s2=ComplexExpand[(a/Ï?)^(1/4) *Exp[-(a t^2)/2]] // Simplify
>>(a^(1/4)*Sqrt[E^(a*t^2)])/(E^(a*t^2)*Pi^(1/4))

\$Assumptions = {a Ïµ Reals, t Ïµ Reals}
s3=ComplexExpand[(a/Ï?)^(1/4) *Exp[-(a t^2)/2]] // Simplify
>>((a^2)^(1/8)*Sqrt[E^(a*t^2)]*(Cos[Arg[a]/4] +
I*Sin[Arg[a]/4]))/(E^(a*t^2)*Pi^(1/4))

s4=Simplify[(a/Ï?)^(1/4) *Exp[-(a t^2)/2]]
>>a^(1/4)/(E^((a*t^2)/2)*Pi^(1/4))

fundamentally you are asking by, using complexexpand, to expand your
given function in a complex way....

>Instead, here is what it does:
>
>In[1]:=
>
>     ComplexExpand[ (a / Pi)^(1/4) Exp[ (-(a t^2)/2 ] ]
>
>Out[2]:=
>
>     (Exp[-a t^2] Sqrt[Exp[a t^2]] (a^2)^(1/8) Cos[Arg[a] / 4]) / Pi^
>(1/4) + i (Exp[-a t^2] Sqrt[Exp[a t^2]] (a^2)^(1/8) Sin[Arg[a] /
>4]) / Pi^(1/4).
>
>I have inserted parentheses in a few places to improve the legibility
>of the expressions.
>
>ComplexExpand treats the variable "a" as complex, but "t" as real.
>This is puzzling to say the least. Moreover, it renders a^(1/4) as
>(a^2)^(1/8), which seems bizarre.
>
>
If you look closely at your expression, the only issue of complexity
occurs with your "a" variable because appears as a radical which may
have complex nature based on where it is defined on the Real number line

>My interest is not in obtaining the correct result, which is easy to
>do. Rather, I bring this up as yet another example of the
>unreliability of ComplexExpand. In case anyone is wondering why I
>would use ComplexExpand on an expression I know to be real, the
>reason is that the expression in question is a factor in a larger
>expression that contains complex variables. Applied to the larger
>expression, ComplexExpand returned an obviously incorrect expansion
>that I traced to the treatment of the example shown above.
>
>I welcome comments and suggestions.
>
>
>Raul Martinez
>
>
>
>
>
>
>
>
Hope this helps

Pratik Desai

--
Pratik Desai